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I definitely don't want to come off as a penny-pincher and haggle over nothing, but what amount of financial aid difference is worth negotiating over?

$10k? $5k? Even $1k?

For my specific circumstances, it's a difference of a little less than $3k for sheer financial aid, though it comes out to be a difference of $7k when it comes to including cost of living (just for the 2015-2016 school year). Worth negotiating?

Yep.

"You miss 100% of the shots you never take"

Any amount.

What is the cost to you of doing so? What is the drawback?

If you are successful, this is the easiest money you will ever make.

Do it just to do it. $5k might not seem like much, but over 13 years with interest (for the longest $5k you owe) that's over $10k. You'll pay it off with your last X number of payments sequentially. You could have bought yourself a Rolex those last few months instead.

Or you could do PAYE, dodge the tax bomb through poverty, and be no worse off.